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Mathematics Exam  >  Mathematics Tests  >  Topic-wise Tests & Solved Examples for Mathematics  >  Differential Equations - 21 - Mathematics MCQ

Differential Equations - 21 - Mathematics MCQ


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20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Differential Equations - 21

Differential Equations - 21 for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation. The Differential Equations - 21 questions and answers have been prepared according to the Mathematics exam syllabus.The Differential Equations - 21 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Differential Equations - 21 below.
Solutions of Differential Equations - 21 questions in English are available as part of our Topic-wise Tests & Solved Examples for Mathematics for Mathematics & Differential Equations - 21 solutions in Hindi for Topic-wise Tests & Solved Examples for Mathematics course. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free. Attempt Differential Equations - 21 | 20 questions in 60 minutes | Mock test for Mathematics preparation | Free important questions MCQ to study Topic-wise Tests & Solved Examples for Mathematics for Mathematics Exam | Download free PDF with solutions
Differential Equations - 21 - Question 1
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The particular integral of the differential eauation (D2 - 4)y  = sin 4x is given by

Detailed Solution for Differential Equations - 21 - Question 1

Proof : The required particular integral is given by

Differential Equations - 21 - Question 2
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The particular integral of the differential equation (D2 + 9)y = cos 2x + sin 2x is given by

Detailed Solution for Differential Equations - 21 - Question 2

Proof : Th required particular integral is given by

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Differential Equations - 21 - Question 3
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The general solution of the differential equation 

Detailed Solution for Differential Equations - 21 - Question 3

The given differential equation is
  ....(i)
Corresponding homogeneous differential equation is
 ....(ii)
The general solution of (i) is given by
y = complementary' function + A particular solution of (i)
or y = General solution of (ii) + A particular solution of (i)
General solution of (ii).
Auxiliary equation is given by 



∴  Required general solution of (i) is given by

Differential Equations - 21 - Question 4
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The particular integral of the differential equation is given by
Detailed Solution for Differential Equations - 21 - Question 4

The given differential equation is
f(D)y = xex,
where f(D) = D2 + 2D + 5
The required Particular Integral is given by


Differential Equations - 21 - Question 5
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The particular integral of the differential equation  sin 2x is given by
Differential Equations - 21 - Question 6
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The particular integral of the differential equation (D2 - 2D + 1)y = xex sin x is given by
Detailed Solution for Differential Equations - 21 - Question 6

The required Particular Integral is given by 

(Note : Im denotes imaginary part of the bracketed quantity)

Differential Equations - 21 - Question 7
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Detailed Solution for Differential Equations - 21 - Question 7

Differential Equations - 21 - Question 8
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A differential equation of the type  where a1, a2, ..., an are constants and X is either a constant or a function of x, is also called
Detailed Solution for Differential Equations - 21 - Question 8

This equation is known as Cauchy Euler equation.

Differential Equations - 21 - Question 9
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The differential equation  can be tranformed into a linear differential equation with constant coefficients by means of the substitution
Detailed Solution for Differential Equations - 21 - Question 9

Proof : The given differential equation is

Now consider the transformation 

   ...(iii)


and so on
Finally we substitute for

into equation (i) and get a differential equation with constant coefficients with y as dependent variable and z as independent variable.

Differential Equations - 21 - Question 10
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Which one of the following differential equations is equivalent to the differential equation 
Detailed Solution for Differential Equations - 21 - Question 10

Hint : Substitute z = log x and using equation (iii) and (iv), we get

Differential Equations - 21 - Question 11
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The initial value problem dy/dx = x, y(0) = 1 has
Detailed Solution for Differential Equations - 21 - Question 11


Differential Equations - 21 - Question 12
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The initial value problem 
Differential Equations - 21 - Question 13
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Which of the following differential equations is exact
Differential Equations - 21 - Question 14
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The first integral of the differential equation  where P0, P1, P2 and P3 are functions of x, is given by
Differential Equations - 21 - Question 15
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The first integral of the differential equation 
Differential Equations - 21 - Question 16
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The first integral of the differential equation 
Differential Equations - 21 - Question 17
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The differential equation 2ydx - (3y - 2x) dy = 0 is
Detailed Solution for Differential Equations - 21 - Question 17



Differential Equations - 21 - Question 18
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The general solution of the differential equation (x + y - 3)dx - (2x + 2y + 1)dy = 0 is
Detailed Solution for Differential Equations - 21 - Question 18


  ......(i)

Differential Equations - 21 - Question 19
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The general solution of the differential equation 
Detailed Solution for Differential Equations - 21 - Question 19



 so differential equation (1) is exact
so its solution is

Differential Equations - 21 - Question 20
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The solution of the differential equation y (2x + y2)dx + x(x + 3y2)dy = 0, is
Detailed Solution for Differential Equations - 21 - Question 20


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